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A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian
1. | Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, 06123 Perugia, Italy |
2. | College of Science, Civil Aviation University of China, Tianjin 300300, China |
3. | Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China |
In this paper, we study an anomalous diffusion model of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we are concerned with the following initial-boundary value problem involving the fractional $p$-Laplacian $\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+M([u]_{s, p}^{p}\text{)}(-\Delta)_{p}^{s}u=f(x, t) & \text{in }\Omega \times {{\mathbb{R}}^{+}}, {{\partial }_{t}}u=\partial u/\partial t, \\ u(x, 0)={{u}_{0}}(x) & \text{in }\Omega, \\ u=0\ & \text{in }{{\mathbb{R}}^{N}}\backslash \Omega, \\\end{array}\text{ }\ \ \right.$ where $[u]_{s, p}$ is the Gagliardo $p$-seminorm of $u$, $Ω\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partialΩ$, $1 < p < N/s$, with $0 < s < 1$, the main Kirchhoff function $M:\mathbb{R}^{ + }_{0} \to \mathbb{R}^{ + }$ is a continuous and nondecreasing function, $(-Δ)_p^s$ is the fractional $p$-Laplacian, $u_0$ is in $L^2(Ω)$ and $f∈ L^2_{\rm loc}(\mathbb{R}^{ + }_0;L^2(Ω))$. Under some appropriate conditions, the well-posedness of solutions for the problem above is studied by employing the sub-differential approach. Finally, the large-time behavior and extinction of solutions are also investigated.
References:
[1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London,, 1975.
![]() ![]() |
[2] |
G. Akagi and K. Matsuura,
Well-posedness and large-time behaviors of solutions for a parabolic equations involving $p(x)$-Laplacian, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications, 8th AIMS Conference.Suppl., 1 (2011), 22-31.
|
[3] |
G. Akagi,
K. Matsuura, Nonlinear diffusion equations driven by the $p(·)$-Laplacian, Nonlinear Differential Equations Appl. NoDEA, 20 (2013), 37-64.
doi: 10.1007/s00030-012-0153-6. |
[4] |
F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo,
A nonlocal $p$-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions, SIAM J. Math. Anal., 40 (2009), 1815-1851.
doi: 10.1137/080720991. |
[5] |
S. Antontsev and S. Shmarev,
Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math.(234), 2010 (), 2633-2645.
doi: 10.1016/j.cam.2010.01.026. |
[6] |
S. Antontsev,
S. Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity, J. Math. Anal. Appl., 361 (2010), 371-391.
doi: 10.1016/j.jmaa.2009.07.019. |
[7] |
D. Applebaum,
Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
|
[8] |
G. Autuori, A. Fiscella and P. Pucci,
Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.
doi: 10.1016/j.na.2015.06.014. |
[9] |
G. Autuori, P. Pucci and M. C. Salvatori,
Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.
doi: 10.1007/s00205-009-0241-x. |
[10] |
H. Brézis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Math Studies, Vol.5 North-Holland, Amsterdam, New York, 1973.
![]() ![]() |
[11] |
L. Caffarelli,
Some nonlinear problems involving non-local diffusions, ICIAM 07-6th International Congress on Industrial and Applied Mathematics, Eur. Math. Soc., Zürich, (2009), 43-56.
doi: 10.4171/056-1/3. |
[12] |
L. Caffarelli,
Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, 7 (2012), 37-52.
doi: 10.1007/978-3-642-25361-4_3. |
[13] |
E. Chasseigne, M. Chaves and J. D. Rossi,
Asymptotic behaviour for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.
doi: 10.1016/j.matpur.2006.04.005. |
[14] |
F. Colasuonno and P. Pucci,
Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974.
doi: 10.1016/j.na.2011.05.073. |
[15] |
C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski,
Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390.
doi: 10.1016/j.jde.2006.12.002. |
[16] |
A. Di Castro, T. Kuusi and G. Palatucci,
Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836.
doi: 10.1016/j.jfa.2014.05.023. |
[17] |
A. Di Castro, T. Kuusi and G. Palatucci,
Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.
doi: 10.1016/j.anihpc.2015.04.003. |
[18] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[19] |
J. M. do'O, O. H. Miyagaki and M. Squassina,
Nonautonomous fractional problems with exponential growth, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1395-1410.
doi: 10.1007/s00030-015-0327-0. |
[20] |
P. Fife,
Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.
|
[21] |
M. Fila,
Boundedness of global solutions of nonlinear diffusion equations, J. Differential Equations, 98 (1992), 226-240.
doi: 10.1016/0022-0396(92)90091-Z. |
[22] |
A. Fiscella, R. Servadei and E. Valdinoci,
Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.
doi: 10.5186/aasfm.2015.4009. |
[23] |
A. Fiscella and E. Valdinoci,
A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.
doi: 10.1016/j.na.2013.08.011. |
[24] |
G. Franzina and G. Palatucci,
Fractional $p$-eigenvalues, Riv. Math. Univ. Parma, 5 (2014), 373-386.
|
[25] |
M. Gobbino,
Quasilinear degenerate parabolic equations of Kirchhoff type, Math. Meth. Appl. Sci., 22 (1999), 375-388.
doi: 10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7. |
[26] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[27] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[28] |
E. Lindgren and P. Lindqvist,
Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.
doi: 10.1007/s00526-013-0600-1. |
[29] |
T. F. Ma,
Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977.
doi: 10.1016/j.na.2005.03.021. |
[30] |
X. Mingqi, G. Molica Bisci, G. H. Tian and B. L. Zhang,
Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 357-374.
doi: 10.1088/0951-7715/29/2/357. |
[31] |
M. Pérez-Llanosa and J. D. Rossi,
Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Anal., 70 (2009), 1629-1640.
doi: 10.1016/j.na.2008.02.076. |
[32] |
P. Pucci and S. Saldi,
Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.
doi: 10.4171/RMI/879. |
[33] |
P. Pucci and J. Serrin,
Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203-214.
doi: 10.1006/jdeq.1998.3477. |
[34] |
P. Pucci, M. Q. Xiang and B. L. Zhang,
Multiple solutions for nonhomogenous Schrodinger-Kirchhoff type equations involving the fractional $p-$Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.
doi: 10.1007/s00526-015-0883-5. |
[35] |
P. Pucci, M. Q. Xiang and B. L. Zhang,
Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.
doi: 10.1515/anona-2015-0102. |
[36] |
R. E. Showalter,
Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs Vol. 49, American Mathematical Society, Providence, RI, 1997, xiv + 278 pp. |
[37] |
J. L. Vázquez,
Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Abel Symp., Springer, Heidelberg, 7 (2012), 271-298.
doi: 10.1007/978-3-642-25361-4_15. |
[38] |
M. Q. Xiang, B. L. Zhang and M. Ferrara,
Existence of solutions for Kirchhoff type problem involving the non-local fractional $p$-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041.
doi: 10.1016/j.jmaa.2014.11.055. |
[39] |
M. Q. Xiang, B. L. Zhang and M. Ferrara,
Multiplicity results for the nonhomogeneous fractional $p$-Kirchhoff equations with concave-convex nonlinearities, Proc. Roy. Soc. A, 471 (2015), 20150034, 14 pp.
doi: 10.1098/rspa.2015.0034. |
[40] |
M. Q. Xiang, B. L. Zhang and V. Rădulescu,
Existence of solutions for perturbed fractional $p$-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413.
doi: 10.1016/j.jde.2015.09.028. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London,, 1975.
![]() ![]() |
[2] |
G. Akagi and K. Matsuura,
Well-posedness and large-time behaviors of solutions for a parabolic equations involving $p(x)$-Laplacian, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications, 8th AIMS Conference.Suppl., 1 (2011), 22-31.
|
[3] |
G. Akagi,
K. Matsuura, Nonlinear diffusion equations driven by the $p(·)$-Laplacian, Nonlinear Differential Equations Appl. NoDEA, 20 (2013), 37-64.
doi: 10.1007/s00030-012-0153-6. |
[4] |
F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo,
A nonlocal $p$-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions, SIAM J. Math. Anal., 40 (2009), 1815-1851.
doi: 10.1137/080720991. |
[5] |
S. Antontsev and S. Shmarev,
Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math.(234), 2010 (), 2633-2645.
doi: 10.1016/j.cam.2010.01.026. |
[6] |
S. Antontsev,
S. Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity, J. Math. Anal. Appl., 361 (2010), 371-391.
doi: 10.1016/j.jmaa.2009.07.019. |
[7] |
D. Applebaum,
Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
|
[8] |
G. Autuori, A. Fiscella and P. Pucci,
Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.
doi: 10.1016/j.na.2015.06.014. |
[9] |
G. Autuori, P. Pucci and M. C. Salvatori,
Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.
doi: 10.1007/s00205-009-0241-x. |
[10] |
H. Brézis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Math Studies, Vol.5 North-Holland, Amsterdam, New York, 1973.
![]() ![]() |
[11] |
L. Caffarelli,
Some nonlinear problems involving non-local diffusions, ICIAM 07-6th International Congress on Industrial and Applied Mathematics, Eur. Math. Soc., Zürich, (2009), 43-56.
doi: 10.4171/056-1/3. |
[12] |
L. Caffarelli,
Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, 7 (2012), 37-52.
doi: 10.1007/978-3-642-25361-4_3. |
[13] |
E. Chasseigne, M. Chaves and J. D. Rossi,
Asymptotic behaviour for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.
doi: 10.1016/j.matpur.2006.04.005. |
[14] |
F. Colasuonno and P. Pucci,
Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974.
doi: 10.1016/j.na.2011.05.073. |
[15] |
C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski,
Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390.
doi: 10.1016/j.jde.2006.12.002. |
[16] |
A. Di Castro, T. Kuusi and G. Palatucci,
Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836.
doi: 10.1016/j.jfa.2014.05.023. |
[17] |
A. Di Castro, T. Kuusi and G. Palatucci,
Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.
doi: 10.1016/j.anihpc.2015.04.003. |
[18] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[19] |
J. M. do'O, O. H. Miyagaki and M. Squassina,
Nonautonomous fractional problems with exponential growth, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1395-1410.
doi: 10.1007/s00030-015-0327-0. |
[20] |
P. Fife,
Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.
|
[21] |
M. Fila,
Boundedness of global solutions of nonlinear diffusion equations, J. Differential Equations, 98 (1992), 226-240.
doi: 10.1016/0022-0396(92)90091-Z. |
[22] |
A. Fiscella, R. Servadei and E. Valdinoci,
Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.
doi: 10.5186/aasfm.2015.4009. |
[23] |
A. Fiscella and E. Valdinoci,
A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.
doi: 10.1016/j.na.2013.08.011. |
[24] |
G. Franzina and G. Palatucci,
Fractional $p$-eigenvalues, Riv. Math. Univ. Parma, 5 (2014), 373-386.
|
[25] |
M. Gobbino,
Quasilinear degenerate parabolic equations of Kirchhoff type, Math. Meth. Appl. Sci., 22 (1999), 375-388.
doi: 10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7. |
[26] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[27] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[28] |
E. Lindgren and P. Lindqvist,
Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.
doi: 10.1007/s00526-013-0600-1. |
[29] |
T. F. Ma,
Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977.
doi: 10.1016/j.na.2005.03.021. |
[30] |
X. Mingqi, G. Molica Bisci, G. H. Tian and B. L. Zhang,
Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 357-374.
doi: 10.1088/0951-7715/29/2/357. |
[31] |
M. Pérez-Llanosa and J. D. Rossi,
Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Anal., 70 (2009), 1629-1640.
doi: 10.1016/j.na.2008.02.076. |
[32] |
P. Pucci and S. Saldi,
Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.
doi: 10.4171/RMI/879. |
[33] |
P. Pucci and J. Serrin,
Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203-214.
doi: 10.1006/jdeq.1998.3477. |
[34] |
P. Pucci, M. Q. Xiang and B. L. Zhang,
Multiple solutions for nonhomogenous Schrodinger-Kirchhoff type equations involving the fractional $p-$Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.
doi: 10.1007/s00526-015-0883-5. |
[35] |
P. Pucci, M. Q. Xiang and B. L. Zhang,
Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.
doi: 10.1515/anona-2015-0102. |
[36] |
R. E. Showalter,
Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs Vol. 49, American Mathematical Society, Providence, RI, 1997, xiv + 278 pp. |
[37] |
J. L. Vázquez,
Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Abel Symp., Springer, Heidelberg, 7 (2012), 271-298.
doi: 10.1007/978-3-642-25361-4_15. |
[38] |
M. Q. Xiang, B. L. Zhang and M. Ferrara,
Existence of solutions for Kirchhoff type problem involving the non-local fractional $p$-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041.
doi: 10.1016/j.jmaa.2014.11.055. |
[39] |
M. Q. Xiang, B. L. Zhang and M. Ferrara,
Multiplicity results for the nonhomogeneous fractional $p$-Kirchhoff equations with concave-convex nonlinearities, Proc. Roy. Soc. A, 471 (2015), 20150034, 14 pp.
doi: 10.1098/rspa.2015.0034. |
[40] |
M. Q. Xiang, B. L. Zhang and V. Rădulescu,
Existence of solutions for perturbed fractional $p$-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413.
doi: 10.1016/j.jde.2015.09.028. |
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